Nonlinear Theory: Optical Mode Analysis

Abstract

The previous chapter dealt with the nonlinear theory in the steady-state regime based on the slowly-varying envelope approximation (SVEA). Most of the time-dependent free-electron laser simulation codes that are in use at the present time deal with an extension of the SVEA in order to solve the wave equation. The time-dependent formulation presented in this chapter is an extension of the SVEA, in which the SVEA is extended by allowing the slowly-varying amplitude to vary in both axial position and time. However, polychromatic formulations using an SVEA where the slowly-varying amplitudes vary only in position can also describe time dependence. This can be accomplished by employing a superposition of modes that are harmonics of an underlying frequency (ω₀). As a result, a time-average over the period 2π/ω₀ will orthogonalize the dynamical equations of the mode amplitudes and allow each harmonic component to be treated separately; hence, this polychromatic approach is equivalent to an explicit Fourier decomposition of the optical field. These two techniques are equivalent; however, in practice the polychromatic approach is least computationally efficient of the two and is not commonly used.

In contrast to the steady-state formulations, a time-dependent formulation is necessary in the simulation of short wavelength free-electron lasers that employ radio frequency linear accelerators (RF linacs) or storage rings. Radio frequency linacs produce high-energy beams with pulse times of the order of 1–10 psec and bunch charges of at most several nano-Coulombs. In X-ray free-electron lasers, the actual bunch charge used is about 250 pC or less. Since the growth rate depends upon the peak current, it is desirable to produce bunches with peak currents of several hundred to several thousand Amperes, and this requires compression of the bunch to sub-picosecond pulse times. For example, the optical field in a free-electron laser operating in the ultraviolet at a wavelength of 300 nm would slip ahead of the electron bunch at a rate of 1 fsec per wiggler period. As a result, the total slippage in a wiggler with a period of 3 cm, and a length of 2 m would be about 67 fsec, which is significant for a sub-picosecond bunch.

In addition to describing the slippage of the optical pulse, time dependence is also needed to study the spectral properties of the optical field such as the temporal coherence, linewidth, sideband production, etc. Furthermore, in contrast to the guided mode analysis used for the steady-state formulation presented in the preceding chapter, the three-dimensional formulations presented in this chapter make use of superpositions of Gaussian optical modes to represent the radiation fields.

The optical field representation described in this chapter is based upon a Gaussian modal superposition of the optical field which treats the x- and y-components of the optical field independently which permits the simulation of a variety of polarization states. This is important because interest has developed in the ability to generate different polarizations – particularly in X-ray FELs that employ a line of relatively short wigglers in sequence. This can be accomplished by a variety of techniques including, but not limited to, (1) alternating the orientations of successive planar wigglers in a long propagation line, (2) using variable polarization wigglers, and (3) using an afterburner wiggler with a different polarization state than those in the preceding sequence.

The particle dynamics are treated by integration of the full, three-dimensional Lorentz force equations as in the development discussed in the preceding chapter. Since short wavelength free-electron lasers typically are based upon high-energy/low-charge electron beams, very long wiggler lines are required. For example, the first X-ray free-electron laser at the Stanford Linear Accelerator Center used 13.5 GeV/250 pC electron beam which necessitated a wiggler line that was about 130 m in length. Since it is impractical to build a single wiggler of that length, the wiggler line consisted of more than 30 individual wigglers of about 3 m in length separated by drift spaces that contained quadrupoles to provide strong focusing of the electron beam. This approach to the electron dynamics permits the simulation to describe the injection and ejection for each wiggler as well as the propagation through the quadrupoles. Further, the slippage of the optical field and relative phase advance of the optical field relative to the electrons is also simulated self-consistently.

Since the SVEA relies upon an average of Maxwell’s equations over the wave period, it is primarily useful for narrow linewidth interactions such as in a free-electron laser, but it is restricted in how short a pulse can be simulated. In order to investigate this short pulse limitation, the formulation/simulation is compared with a particle-in-cell simulation which has no such limitation. As demonstrated in this chapter, the SVEA is able to reliably simulate pulses as short as the cooperation length.

As in the preceding chapter, comparisons with various experiments are described that validate the formulation.